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synthetic division

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2 Answers

x^4-6x^3+x^2+24x-20
 
let's try 2
 
 
2           1      -6      1      24      -20
                      2      -8    -14       20
                     
             1      -4     -7      10        0
 now divide by (x-2)
 
                x^3-4x^2-7x+10
             __________________
    x-2  / x^4-6x^3+x^2+24x-20
          /  x^4-2x^3
            ________
                    -4x^3+x^2
                    -4x^3+8x^2
                   ___________
                              -7x^2+24x
                              -7x^2+14x
                             __________
                                          10x-20
                                          10x-20
                                          _______
                                                   0
 
using x^3-4x^2-7x+10 try 1 using synthetic division
 
 
      1                  1      -4      -7      10
                                   1       -3     -10
 
                           1     -3      -10      0
 
 
divide x^3-4x^2-7x+10 by x-1
     
              x^2-3x-10
             __________________
     x-1 / x^3-4x^2-7x+10 
         /   x^3-x^2
           _________
                    -3x^2-7x
                    -3x^2+3x
                    ________
                             -10x+10
                             -10x+10
                             _______
                                        0
factor x^2-3x-10 into (x-5)(x+2)
 answer (x-2)(x-1)(x-5)(x+2) = x^4-6x^3+x^2+24x-20                       

Comments

Why both synthetic division and ordinary division?  They are, for these divisions, equivalent.
You are correct, Michael. By actually dividing you see that the coefficients of the quotient are equal to the results of the synthetic division directly above. The actual division shows that no mistakes were made in the synthetic division process and the number used is really a zero. The student sees the relationship between synthetic division and actual division; that they are, as you said, equivalent.

Comment

Using synthetic division,
2|1 -6 1 24 -20
......2 -8 -14 20---------------------
..1 -4 -7 10|0
 
In the same way, you can see k = -1 and k = 0 are not zeros.
f(-1) = -36
f(0) = -20